Connecting mirror symmetry in 3D and 2D via localization

We explicitly apply localization results to study the interpolation between three- and two-dimensional mirror symmetries for Abelian gauge theories with four supercharges. We first use the ellipsoid S-b(3) partition functions to verify the mirror symmetry between a pair of general three- dimensional N = 2 Abelian Chern-Simons quiver gauge theories. These expressions readily factorize into holomorphic blocks and their antiholomorphic copies, so we can also obtain the partition functions on S-1 x S-2 via fusion procedure. We then demonstrate S1 x S2 partition functions for the three- dimensional Abelian gauge theories can be dimensionally reduced to the S2 partition functions of N = ( 2, 2) GLSM and Landau-Ginzburg model for the corresponding two-dimensional mirror pair, as anticipated previously in M. Aganagic et al., J. High Energy Phys. 0107, 022 (2001). We also comment on the analogous interpolation for the non-Abelian gauge theories and compute the K-theory vortex partition function for a simple limit to verify the prediction from holomorphic block.